Tits' vision was then popularized by Soulé who came up in [1]} with
a first tentative definition of affine varieties over \(\mathbb {F}_1\) . Later on,
other constructions were proposed and the subject has attracted more and
more attention for the last two decades [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}; see [10]} for a recent review on this topic.
Nowadays, the theory of \(\mathbb {F}_1\)  is not well estalished yet.
However, several definitions for the category of schemes over \(\mathbb {F}_1\) 
have been proposed over the years and significant progress towards Tits'
initial dream have been realized. Besides, geometry over \(\mathbb {F}_1\)  have
been extended to new contexts and have nowadays close interactions with
Arakelov geometry and \(p\) -adic geometry.
In particular, Bambozzi, Ben-Bassat and Kremnizer introduced
in [9]} analytic geometry over \(\mathbb {F}_1\) ; they managed notably
to construct a model of the Fargues-Fontaine curve [12]} in their
theory. Up to our knowledge, this was the first connexion between the
theory of Galois representations (incarnated here by \(p\) -adic Hodge
theory) and the field with one element.
